Problem: Solve for $x$ : $ 6|x - 1| + 1 = 1|x - 1| + 6 $
Solution: Subtract $ {1|x - 1|} $ from both sides: $ \begin{eqnarray} 6|x - 1| + 1 &=& 1|x - 1| + 6 \\ \\ { - 1|x - 1|} && { - 1|x - 1|} \\ \\ 5|x - 1| + 1 &=& 6 \end{eqnarray} $ Subtract ${1}$ from both sides: $ \begin{eqnarray} 5|x - 1| + 1 &=& 6 \\ \\ { - 1} &=& { - 1} \\ \\ 5|x - 1| &=& 5 \end{eqnarray} $ Divide both sides by ${5}$ $ \dfrac{5|x - 1|} {{5}} = \dfrac{5} {{5}} $ Simplify: $ |x - 1| = 1$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 1 = -1 $ or $ x - 1 = 1 $ Solve for the solution where $x - 1$ is negative: $ x - 1 = -1 $ Add ${1}$ to both sides: $ \begin{eqnarray} x - 1 &=& -1 \\ \\ {+ 1} && {+ 1} \\ \\ x &=& -1 + 1 \end{eqnarray} $ $ x = 0 $ Then calculate the solution where $x - 1$ is positive: $ x - 1 = 1 $ Add ${1}$ to both sides: $ \begin{eqnarray} x - 1 &=& 1 \\ \\ {+ 1} && {+ 1} \\ \\ x &=& 1 + 1 \end{eqnarray} $ $ x = 2 $ Thus, the correct answer is $x = 0 $ or $x = 2 $.